I am trying to figure out how to quantify the change in price on a bond for a change in credit risk. I'm not even sure how to quantify a change in credit risk, but I'm thinking possibly something related to either the debt/equity rating of a corporate bond or a change in the credit rating. If there is a better way, please do include it in your answer.

So, my question is how to determine the price change from a change in the credit risk for a bond? Is there a way to quantify this in any remotely accurate way?
If so, how?

What you're looking for looks to be more in the realm of a mathematical model (specific to the company's size, available liquidity, and industry). Credit Risk Pricing Models may provide a decent overview of how to build such a model.

Unfortunately duration/convexity will only help you capture the interest rate risk on your bonds, and not any of the idiosyncratic events such as credit downgrades.